ON THE PROPERTIES OF A KIND OF RANDOM MATRICES
Starting from matrices whose columns generate an isotropic subspace and the work done by Qing-You in [8], important properties of a kind of random symplectic matrix are presented. We show that:
(1) it can be transformed into Jordan canonical form by a similar orthogonal transformation,
(2) it has a particular Schur canonical form, and
(3) its condition number is a constant and is the same as that of the matrix studied in [8], numerical examples are given to confirm our theoretical results.
Keywords and phrases: symplectic matrix, perturbations, Schur form, Jordan form.
Received: January 6, 2023; Accepted: February 17, 2023; Published: July 24, 2023
How to cite this article: Traoré G. Y. Arouna and Haudié Jean Stéphane Inkpé, On the properties of a kind of random matrices, Far East Journal of Dynamical Systems 36(1) (2023), 93-104. http://dx.doi.org/10.17654/0972111823004
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:[1] R. Abraham and J. Marsden, Foundations of Mechanics, Second ed., Addison-Wesley, Reading, 1978.[2] V. I. Arnold, Mathematical Methods in Classical Mechanics, Springer-Verlag, Berlin, 1978.[3] Traoré G. Y. Arouna, M. Dosso and J. C. Koua Brou, On a perturbation theory of Hamiltonian systems with periodic coefficients, International Journal of Numerical Methods and Applications 17(2) (2018), 47-89.[4] M. Dosso, Traoré G. Y. Arouna and J.-C. Koua Brou, On rank one perturbation of Hamiltonian system with periodic coefficients, WSEAS Trans. Math. 15 (2016), 502-510.[5] G. Freiling, V. Mehrmann and H. Xu, Existence, uniqueness and parametrization of Lagrangian invariant subspaces, SIAM J. Matrix Anal. Appl. 23(4) (2002), 1045-1069.[6] F. Poloni and N. Strabić, Principal pivot transforms of quasidefinite matrix and semidefinite Lagrangian subspaces, Journal of Linear Algebra 31 (2016), 200-231.[7] V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, Vols. 1 & 2, Wiley, New York, 1975.[8] Yan Qing-You, The properties of a kind of random symplectic matrices, Appl. Math. Mech. 23(5) (2002), 590-596.
